The organization of US presidential elections makes them potentially vulnerable to the “voting paradoxes” identified by social choice theorists but rarely documented in real-world elections. Using polling data from the 2016 Republican presidential primaries we identify two possible cases: Early in the pre-primary (2015) a cyclical majority may have existed in Republican voters’ preferences between Bush, Cruz and Walker. Furthermore, later polling data (January-March 2016) suggests that while Trump (who achieved less than 50% of the total Republican primary vote) was the Plurality Winner, he could have been beaten in pairwise contests by at least one other candidate and may have been the Condorcet Loser. The cases confirm the empirical relevance of the theoretical voting paradoxes and the importance of voting procedures.